Solve each problem. Alcohol Mixture Barak wishes to strengthen a mixture that is 10% alcohol to one that is 30% alcohol. How much pure alcohol should he add to 12 L of the 10% mixture?

Verified step by step guidance

1. Identify the variables: Let \(x\) be the amount of pure alcohol (100% alcohol) to add to the mixture.
2. Determine the amount of alcohol in the original mixture: Since the mixture is 10% alcohol and there are 12 liters, the amount of alcohol is \$0.10 \times 12$ liters.
3. Set up the equation for the final mixture: After adding \(x\) liters of pure alcohol, the total volume becomes \$12 + x\( liters, and the total amount of alcohol becomes \)0.10 \times 12 + x$ liters.
4. Write the equation representing the desired concentration of 30% alcohol in the new mixture: \(\frac{0.10 \times 12 + x}{12 + x} = 0.30\).
5. Solve the equation for \(x\) to find how much pure alcohol should be added.

Key Concepts

Concentration and Percentage

Concentration refers to the amount of a substance (like alcohol) present in a mixture, often expressed as a percentage. Understanding how to interpret and manipulate these percentages is essential for solving mixture problems involving different concentrations.

Mixture Problems and Setting Up Equations

Mixture problems require setting up equations based on the total amount and concentration of substances before and after mixing. By expressing the quantities and concentrations algebraically, you can solve for unknown amounts, such as how much pure alcohol to add.

Solving Linear Equations

Once the mixture problem is translated into an equation, solving linear equations is necessary to find the unknown variable. This involves isolating the variable on one side and performing arithmetic operations to determine the solution.